Decentralized Supervisory Control with Communicating Supervisors Based on Top-Down Coordination Control
Jan Komenda and Tom´aˇs Masopust
Abstract— In this paper we present a new approach to de- centralized supervisory control of large automata with commu- nicating supervisors. We first generalize the recently developed top-down architecture of multilevel coordination control with a hierarchical structure of groups of subsystems, their respective coordinators and supervisors. Namely, in the case where the equivalent conditions for achieving a specification language fail to be satisfied, we propose sufficient conditions for a dis- tributed computation of the supremal achievable sublanguage.
We then apply the obtained constructive results of multilevel coordination control to decentralized supervisory control with communication, where local supervisors of subsystems within a group communicate with each other via the coordinator of the group. Our approach is illustrated by an example.
I. INTRODUCTION
Decentralized supervisory control has been introduced in [11] aiming at decreasing the computational complexity of supervisory control of large automata without a product structure based on a single (centralized) supervisor. As it turned out that coobservability of a specification language, the necessary condition to achieve the specification in de- centralized supervisory control, is too strong, two different approaches have been proposed. The first one is based on communication between supervisors to achieve coobservabil- ity with respect to observable alphabets enriched by commu- nicated event occurrences, cf. [1] and [9]. The second one consists in proposing new and more general decentralized su- pervisory control architectures that lead to weaker notions of coobservability. The original notion of coobservability [11]
has been called conjunctive and permissive (C & P), while an alternative architecture called disjunctive and antipermissive (D & A) has been proposed in [13] together with their com- bination. Among weaker concepts of coobservability, one should cite a general architecture combining both architec- tures [13] or even more general architectures with possible several levels of inferencing leading to conditional versions of coobservability [14].
Coordination control [8] can be seen as a trade-off between a purely local (modular) control synthesis that often fails in achieving sufficiently permissive supervisors and leads to blocking, and a global control synthesis that is too expensive from the computational complexity viewpoint. We have extended coordination control to the multilevel setting with a hierarchical structure of groups of subsystems, their
J. Komenda (komenda@math.cas.cz) is with the Institute of Mathematics, Academy of Sci. of the Czech Republic, ˇZiˇzkova 22, 616 62 Brno, Czech Republic. T. Masopust (masopust@math.cas.cz) is with Technische Universit¨at Dresden, Germany.
respective coordinators, and supervisors in [7]. The top- down approach proposed therein along with the correspond- ing notions of conditional decomposability and conditional controllability enables to compute the supervisors only at the lowest level. Unlike centralized coordination, supervisors of subsystems communicate only within the groups on the same level of hierarchy via the group coordinators located on the next upper level of hierarchy.
In [5], we have proposed a generic approach that consists in applying the results of coordination control of automata with a synchronous-product structure to decentralized control of automata without an explicit product structure. It is based on the over-approximation of the automaton without a product structure by a product of its natural projections to the alphabet of local supervisors. However, a communication structure is not interesting, because all local supervisors communicate with each other via a coordinator.
In this paper we benefit from recent results of multilevel coordination control. First, earlier existential results [7] are extended to a construction procedure that computes the least restrictive solution for the top-down architecture of multi- level coordination control. These results are then applied to the original decentralized control problem, which is solved using an underlying communication scheme, where local supervisors of subsystems within a group communicate with each other via a coordinator of the group. The optimal solution obtained by coordination control leads to a (possibly non-optimal) solution of the original decentralized control problem.
II. EXISTENTIAL RESULTS OF MULTILEVEL COORDINATION CONTROL
Basic notions and notational conventions are first recalled.
The free monoid of strings over an alphabetAis denoted by A∗. A language is a subset ofA∗. The prefix closure of a lan- guageL⊆A∗is the set of all prefixes of all its strings and is denoted byL={w∈A∗|there existsv∈A∗ such thatwv∈ L};L is prefix-closed ifL=L. All languages considered in this paper are assumed to be prefix-closed.
Ageneratoris a quadrupleG= (Q,A,f,q0)consisting of a finite set ofstates Q, a finitealphabet A, apartial transition function f :Q×A→Q, and an initial state q0∈Q. The function f can be extended in the standard way to strings, i.e.
f:Q×A∗→Q. We recall thatL(G) ={s∈A∗|f(q0,s)∈Q}
is called thegenerated language ofG.
A controlled generator over an alphabet A is a triple (G,Ac,Γ), whereGis a generator overA,Ac⊆Ais a set of controllable events,Au=A\Ac is the set of uncontrollable
events, andΓ={γ⊆A|Au⊆γ}is theset of control patterns.
A supervisor for a controlled generator (G,Ac,Γ)is a map S:L(G)→Γ. The closed-loop system associated with the controlled generator(G,Ac,Γ)and the supervisorSis defined as the minimal languageL(S/G)such that ε∈L(S/G)and, for anys∈L(S/G)withsa∈L(G)anda∈S(s),sa belongs toL(S/G).
A language K⊆A∗ is controllable with respect toL and Au ifKAu∩L⊆K.
Aprojection P:A∗→B∗, forB⊆A, is a homomorphism defined asP(a) =ε, fora∈A\B, andP(a) =a, fora∈B.
Theinverse imageofP, denoted byP−1:B∗→2A∗, is defined as P−1(w) ={s∈A∗|P(s) =w}. These definitions can be extended to languages. For alphabetsAi,Aj,A`⊆A, we use P`i+j to denote the projection from(Ai∪Aj)∗ toA∗`. If Ai∪ Aj=A, we simply writeP`. Moreover,Ai,u=Ai∩Audenotes the set of locally uncontrollable events. For a generator G and a projectionP,P(G)denotes the minimal generator such that L(P(G)) =P(L(G)). The reader is referred to [2] for a construction.
LetGbe a generator over an alphabetA. Given a specifi- cationK⊆L(G), the aim of supervisory control is to find a supervisorSsuch thatL(S/G) =K. Such a supervisor exists if and only ifK is controllable with respect toL(G)andAu, see [2].
The synchronous product of languages Li⊆A∗i, for i= 1, . . . ,n, is defined askni=1Li=∩ni=1Pi−1(Li)⊆A∗, whereA=
∪ni=1Ai, and Pi:A∗→A∗i are projections to local alphabets.
The corresponding synchronous product of generators Gi (see [2] for definition) satisfies L(kni=1Gi) =kni=1L(Gi).
A projectionQ:A∗→B∗ is anL-observerfor a language L⊆A∗ if, for every t∈Q(L) and s∈L, Q(s) is a prefix of t implies that there exists u∈A∗ such that su∈L and Q(su) =t, cf. [12].
We need the following obvious lemma and results below.
Lemma 1: For any language L⊆A∗ and projections P1: A∗→B∗1 and P2:A∗→B∗2 with B2⊆B1⊆A, it holds that P1(L)kP2(L) =P1(L).
Lemma 2 (Lemma 4.3 in [3]): Let K⊆Lbe controllable with respect toLandAu. If the natural projectionP:A∗→A∗o is an L-observer and OCC for L, then P(K) is controllable with respect toP(L) andAu∩Ao.
Lemma 3 (Proposition 4.6 in [3]): For i = 1, . . . ,n, let Ki⊆Li be controllable with respect toLi⊆A∗i andAi,u, then kni=1Ki is controllable with respect tokni=1Li and∪ni=1Ai,u.
Now we recall the basic notions of coordination con- trol [8]. A language K over ∪ni=1Ai is conditionally decom- posable with respect to alphabets (Ai)ni=1 and Ak, where
∪i6=j1≤i,j≤n(Ai∩Aj)⊆Ak⊆ ∪ni=1Aj, if K=kni=1Pi+k(K),
for projectionsPi+k from∪nj=1Aj toAi∪Ak, fori=1, . . . ,n.
Alphabet Ak is referred to as a coordinator alphabet and satisfies the conditional independence property, that is, Ak includes all shared events: ∪i6=j1≤i,j≤n(Ai∩Aj)⊆Ak. This has the following well known impact.
Lemma 4 ([3]): LetPk:A∗→A∗k be a projection, and let Li be a language overAi, for i=1, . . . ,n, and∪i6=j1≤i,j≤n(Ai∩ Aj)⊆Ak. ThenPk(kni=1Li) =kni=1Pk(Li).
The idea of coordination control is to first construct a su- pervisorSksuch that the closed-loop systemL(Sk/Gk)satis- fies the ”coordinator part” of the specification given byPk(K) and then local supervisorsSi for plantsGik(Sk/Gk), for i= 1, . . . ,n, such that the closed-loop systemL(Si/[Gik(Sk/Gk)]) satisfies the corresponding part of the specification given by Pi+k(K). Conditional controllability along with conditional decomposability form an equivalent condition for a language to be achieved by the closed-loop system within our coordi- nation control architecture, see below.
A language K⊆L(G1kG2k. . .kGnkGk) is conditionally controllable for generatorsG1,G2, . . . ,Gnand a coordinator Gk and uncontrollable alphabetsA1,u,A2,u, . . . ,An,u, andAk,u if (1) Pk(K) is controllable with respect to L(Gk) andAk,u and (2)Pi+k(K)is controllable with respect toL(Gi)kPk(K) andAi+k,u=Ai+k,u= (Ai∪Ak)∩Au, fori=1,2, . . . ,n.
Recall that every conditionally controllable and condition- ally decomposable language is controllable [8]. The main existential result of [8] states that for a specificationK⊆A∗ that is conditionally decomposable, there exist supervisors S1,S2, . . . ,Sn, andSk such that kni=1L(Si/[Gik(Sk/Gk)]) =K if and only ifK is conditionally controllable.
III. CONSTRUCTIVE RESULTS OF TWO-LEVEL COORDINATION CONTROL
In this section we assume that G=G1kG2k. . .kGn and that the subsystems are organized intomgroupsIj, for j= 1,2, . . . ,m. The notation
AIr=[
i∈IrAi
is used in the paper. Here PIr denotes the projection PIr : A∗→A∗I
r. The notation for a projection to extended group events PIr+k:A∗→(Ak∪AIr)∗ should be self-explanatory.
We have introduced the corresponding notion of conditional decomposability in [7].
Definition 5 (Two-level conditional decomposability):
A languageK⊆A∗is calledtwo-level conditionally decom- posable with respect to alphabets A1,A2, . . . ,An, high-level coordinator alphabetAk, and low-level coordinator alphabets Ak1,Ak2, . . .Akm if
K=kmr=1PIr+k(K) and PIr+k(K) =kj∈IrPj+kr+k(K) forr=1,2, . . . ,m.
Remark 6: Unlike the original approach in [7] we propose the following simplification. Instead of using both low-level coordinator alphabetsAkj, j=1,2, . . . ,m, and high-level co- ordinator alphabetAk, we will only use low-level coordinator alphabetsAkj, j=1,2, . . . ,m. We recall thatAkcontains only events shared between different groups of subsystems, that is,Ak⊇Sk6=`k,`∈{1,2,...,m}(AIk∩AI`), which is typically a much smaller set than the set of shared events (between two or more subsystems). Thus, for j=1,2, . . . ,m, we define new low-level coordinator alphabets Akj :=Ak∪Akj by putting
into the alphabets of group coordinators Akj also events from the global coordinator set (if this is nonempty). This is possible, because only prefix-closed languages are used, which means that no high-level coordinators for nonblocking are needed. We recall that in the prefix-closed case the coordinators are actually determined by the corresponding alphabets from Definition 5 as a projection of the plant to these alphabets. This simplification is used because this paper is technically involved.
Problem 7 (Two-level coordination control problem):
Let generators G1, G2,. . . , Gn be over the alphabets A1, A2,. . . , An, respectively, and consider the modular system as their synchronous product G =G1kG2k. . .kGn along with the two-level hierarchical structure of subsystems organized into groups Ij, for j =1,2, . . . ,m and m≤n, on the low level. The synchronous products ki∈Ij Gi, for j =1,2, . . . ,m, then represent the m high-level systems.
Coordinators Gkj are associated to groups of subsystems {Gi|i∈Ij}, for j=1,2, . . . ,m. The two-level coordination control problem consists in synthesizing supervisors Si for any low-level systemsGi, fori=1,2, . . . ,n, and higher-level supervisors Skj supervising the group coordinator Gkj, for j =1,2, . . . ,m, such that the specification is met by the closed-loop system. Then the two-level coordinated and supervised closed-loop system is
kmj=1ki∈Ij L(Si/[Gik(Skj/Gkj)]). / For a specificationK, the coordinatorGkj of the j-th group of subsystems{Gi|i∈Ij} is computed as follows.
1) Set Akj =Sk6=`k,`∈I
j(Ak∩A`) to be the set of all shared events of systems from the groupIj.
2) ExtendAkj so that PIj+k(K) is conditional decompos- able with respect to(Ai)i∈Ij andAkj, for instance using a method described in [6].
3) Let coordinator Gkj=kni=1Pkj(Gi).
With the definition thatAk⊆Akj described in Remark 6, we can simplifyL(Gk)kL(Gkj)of [7] toL(Gkj). Indeed, by our choice of coordinators L(Gk)kL(Gkj) =Pk(L)kPkj(L) = Pkj(L) =L(Gkj), where L=kni=1.
Therefore, instead of both low-level coordinatorsGkj, for j=1,2, . . . ,m, for subsystems belonging to individual groups {Gi|i∈Ij} and high-level coordinators Gk that coordinate the different groups, we are using only low-level (group) coordinatorsGkj, but over larger alphabets compared to [7].
Since the only known condition ensuring that the projected generator is smaller than the original one is the observer property [12] we might need to further extend alphabetsAkj so that projectionPkj is anL(Gi)-observer, for alli∈Ij.
We assume that the specification is prefix-closed, hence the blocking issue is not considered in this paper. Blocking can be handled using coordinators for nonblockingness studied in [8].
The key concept is the following.
Definition 8 ([7]): Consider the setting and notations of Problem 7, and let Gk be a coordinator. A language K⊆ L(kni=1Gi)istwo-level conditionally controllablewith respect
to generators G1,G2, . . . ,Gn, local alphabets A1,A2, . . . ,An, low-level coordinator alphabets Ak1,Ak2, . . .Akm, and uncon- trollable alphabetAuif
1) Pkj(K)is controllable with respect toL(Gkj)andAkj,u, 2) for j=1,2, . . . ,m and i∈Ij, Pi+kj(K) is controllable
with respect toL(Gi)kPkj(K)andAi+kj,u.
Note that we have simplified the original version of two-level conditional controllability from [7] by replacing the composition L(Gk)kL(Gkj) by L(Gkj) as discussed in Remark 6. For a future reference we will say thatK is two- level conditionally controllable with respect toGi, fori∈Il, andGkl, i.e., several items are omitted (but should be clear from the problem formulation).
The following lemma shows how to construct a two-level conditional controllable language as the synchronous com- position of conditionally controllable languages for groups.
Lemma 9: For all l=1,2, . . . ,m, let the languages Ml⊆ A∗I
l be conditionally controllable with respect toGi, fori∈ Il, andGkl, and conditionally decomposable with respect to alphabets Ai, fori∈Il, and Akl, and Ak`⊇Ak⊇Sk6=`(AIk∩ AI`).
If for alll=1,2, . . . ,m,Pkkl is aLkl-observer and OCC for Pkl(Ml), thenkml=1Ml is two-level conditionally controllable with respectGi, fori∈Il, andGkl, for l=1,2, . . . ,m.
The main existential result of multilevel coordination control is now recalled from [7].
Theorem 10: Consider the setting of Problem 7 (in par- ticular K is two-level conditionally decomposable with respect to local alphabets A1,A2, . . . ,An, high-level coor- dinator alphabet Ak, and low-level coordinator alphabets Ak1,Ak2, . . .Akm). There exist supervisors Si, for i∈Ij, for low-level systems within any group of low-level systems {Gi |i∈Ij}, for j =1,2, . . . ,m, and supervisors Skj, for
j=1,2, . . . ,m, for low-level coordinators such that
kmj=1ki∈Ij L(Si/Gik(Skj/Gkj)) =K (1) if and only if K is two-level conditionally controllable as defined in Definition 8.
In the last section we have recalled two-level coordination control framework with the main existential result. A natural question is what to do if the specification fails to satisfy the necessary and sufficient conditions for being achievable. We recall from [8] that in the case specification K fails to be conditionally controllable, the supremal conditionally con- trollable sublanguage always exists and can be computed in a distributive way. First, we show that two-level conditional controllability is closed under language unions as well.
Theorem 11: Two-level conditional controllability is closed under language unions, that is, supremal two-level conditional controllable languages always exist.
By Theorem 11, the supremal two-level conditional con- trollable sublanguage of a specification K with respect to Ak1,Ak2, . . .Akm, and Au, denoted by sup 2cC(K,L,Ai+kj), always exists. Below we propose a procedure to compute the supremal two-level conditional controllable sublanguage sup 2cC(K,L,Ai+kj).
Similarly as in the centralized coordination we introduce the following notation. For all j=1,2, . . . ,mandi∈Ij,
sup Ck
j=sup C(Pkj(K),L(Gkj),Akj,u) sup Ci+k
j=sup C(Pi+kj(K),L(Gi)ksup Ck
j,Ai+kj,u) (2)
where sup C(K,L,Au)denotes the supremal controllable sub- language ofK with respect toL andAu, see [2].
Similarly as in the centralized coordination, the following inclusion always holds true.
Lemma 12: For all j=1,2, . . . ,m and for all i∈Ij, we have that Pki+kj
j (sup Ci+k
j)⊆sup Ck
j.
Transitivity of controllability is needed later.
Lemma 13 ([8]): Let K ⊆L⊆M be languages over A such thatKis controllable with respect toLandAu, andLis controllable with respect toMandAu. ThenKis controllable with respect toM andAu.
The main constructive result follows.
Theorem 14: Consider Problem 7 and languages defined in (2). If ∩i∈IjPki+kj
j (sup Ci+k
j) is controllable with respect toL(Gkj)and Akj,u, and if for all j=1,2, . . . ,m, Pkkj is an Lkj-observer and OCC for Lkj, then sup 2cC(K,L,Ai+kj) = kmj=1ki∈Ijsup Ci+k
j.
Note that controllability of ∩i∈IjPki+kj
j (sup Ci+k
j)with re- spect to L(Gkj) and Akj,u for all j=1,2, . . . ,m, is not a suitable condition for verification. Clearly, for our prefix- closed languages, controllability of Pki+kj
j (sup Ci+k
j) with
respect to L(Gkj) and Akj,u, for j=1,2, . . . ,m and i∈Ij, implies it. Moreover, two stronger checkable conditions are provided below.
It is easy to see that the equality in Lemma 12 implies the sufficient condition of Theorem 14. Indeed, if for all
j=1,2, . . . ,mand for alli∈Ij, we havePki+kj
j (sup Ci+k
j)⊆
sup Ck
j, then in particular Pki+kj
j (sup Ci+k
j) is controllable with respect to L(Gkj) and Akj,u. Hence, for all j = 1,2, . . . ,m,∩i∈IjPki+kj
j (sup Ci+k
j)is controllable with respect toL(Gkj)andAkj,u, which proves the following result.
Corollary 15: Consider the setting of Problem 7 and the languages defined in (2). If for all j =1,2, . . . ,m, Pkkj is an Lkj-observer and OCC for Lkj, and for all i∈ Ij, Pki+kj
j (sup Ci+k
j) =sup Ck
j, then sup 2cC(K,L,Ai+kj) = kmj=1ki∈Ij sup Ci+k
j.
There is yet another sufficient condition that guarantees the controllability requirement in Theorem 14. Namely, local control consistency and observer properties (that are checkable by well-known methods).
Lemma 16: For all j=1,2, . . . ,m and i∈Ij, let Pki+kj
j
be an(Pii+kj)−1L(Gi)-observer and OCC for(Pii+kj)−1L(Gi).
Then, for all j=1,2, . . . ,m,∩i∈IjPki+kj
j (sup Ci+k
j)is control- lable with respect toL(Gkj)andAkj,u.
We point out that even without the above conditions, kmj=1ki∈Ijsup Ci+k
j is controllable with respect to L, but we cannot guarantee maximal permissiveness with respect to the two-level coordination control architecture.
IV. DECENTRALIZED SUPERVISORY CONTROL WITH COMMUNICATION
In this section, constructive results of the top-down coordi- nation are applied to decentralized supervisory control with communicating supervisors. To avoid any confusion we use systematicallyΣto denote alphabets in decentralized control, while notation A is reserved for alphabets in coordination control. Decentralized supervisory control is now briefly recalled.
A. Decentralized Supervisory Control Problem
Decentralized supervisory control differs from modular or coordination control, because the global system is a large automaton without a product structure. In decentralized supervisory control sensing and actuating capabilities are distributed among local supervisors (Si)ni=1 such that each Si observes a subsetΣo,i⊆Σand based on its observation it can disable its controllable eventsΣc,i. Projections to locally observable events are denoted by Pi :Σ∗→Σ∗o,i. We use the notation Σc=∪ni=1Σc,i, Σo=∪ni=1Σo,i, Σu=Σ\Σc, and Σuo=Σ\Σo.
Formally, a local supervisorSifor a generatorGis defined as a mappingSi:Pi(L(G))→Γ, whereΓi={γ⊆Σ|γ⊇(Σ\ Σc,i)}is the set of local control patterns, andSi(s)represents the set of locally enabled events whenSi observes a string s∈Σ∗o,i. Then the permissive local supervisor law isSi(s) = (Σ\Σc,i)∪{a∈Σc,i| ∃s0∈K withPi(s0) =Pi(s)ands0a∈K}.
The global control law S is given by conjunction of local ones: for w∈Σ∗o,i, S(w) =∩ni=1Si(Pi(w)), This is why this control architecture is nowadays referred to as conjunctive and permissive.
Definition 17: K⊆L isC&P coobservablewith respect to L=L(G) and (Σo,i)ni=1 if for all s∈K, a∈Σc, and sa∈L\K, there exists i∈ {1,2, . . . ,n} such that a∈Σc,i and(Pi−1(Pi(s)){a} ∩K=/0.
C&P coobservability can be interpreted in the following way: if we exit from the specification by an event a, then there must exists at least one local supervisor that can control this event (a∈Σc,i) and can disableaunambiguously, i.e., all lookalike (forSi) strings exit the specification as well. Recall that C&P coobservability can be decided in polynomial time [10] in the number of states of the specification and system (but in exponential time in the number of local supervisors).
Since the counterpart of C & P coobservability, called D & A coobservability, is not studied in this paper, C & P coobservability will be referred to as coobservability.
It has been proved in [11] that controllability and coob- servability are the necessary and sufficient conditions to achieve a specification as the resulting closed-loop language.
For languages that fail to satisfy these conditions it is impor- tant to compute a controllable and coobservable sublanguage.
We first recall that decomposability is strongly related to coobservability. A languageK isdecomposablewith respect to alphabets(Σi)ni=1andLifK=kni=1Pi(K)∩L. In the special caseL=Σ∗ decomposability is called separability [4], i.e.,
K is separable with respect to alphabets (Σi)ni=1 if K = kni=1Pi(K).
Now we are ready to recall that (under the alphabet condition that will be shown non-restrictive in the following subsection) separability implies coobservability.
Theorem 18 ([5]): Assume that Σo,i∩Σc⊆Σc,i, for i= 1,2, . . . ,n. If K is separable with respect to (Σo,i)ni=1, then K∩Lis coobservable with respect to (Σo,i)ni=1 andL.
B. Construction of controllable and coobservable sublan- guages based on two-level coordination control
Now we show how the new constructive results of the top-down approach to multilevel coordination control from Section III can be used to compute sublanguages that are by construction controllable and coobservable.
The idea is that owing to our results of coordination control, the supremal conditionally controllable sublanguage (that is in particular controllable) can be computed as a synchronous product of languages over alphabets enriched by communicated events from group coordinators. Hence the resulting language is by construction not only controllable, but in view of Theorem 18 also coobservable. Moreover, due to the multilevel coordination, coordinator events are not communicated among all subsystems, which was our first approach presented in [5] based on the centralized coordination. Roughly speaking, coordinator events from the centralized coordination are distributed to group coordinator events, and only these are to be communicated among the subsystems belonging to the group.
Recall at this point that alphabets in decentralized control are denoted byΣ, while alphabets in coordination control are denoted by A.
It follows from Theorem 14 and its consequences that the supremal two-level conditionally-controllable sublanguage is decomposable with respect to alphabets(Ai+kj)j=1,2,...,m,i∈Ij. Note that conditional decomposability with respect to alphabets(Σo,i)ni=1andΣk such that∪i6=j(Σi∩Σj)⊆Σk, i.e.,
K=kni=1Pi+k(K),
is nothing else but separability of K with respect to (Σo,i∪ Σk)ni=1. An important feature of separability with respect to the alphabets of this form with intersection between all pairs of alphabets being equal toΣk is that it can be checked in polynomial time in the number of local agents [6].
We recall that there always existsΣkthat makes language K conditionally decomposable with respect to (Σi)ni=1 and Σk, cf. [6]. Moreover, K is conditionally decomposable if and only if there existMi⊆Σ∗i+k such thatK=kni=1Mi.
Consider now the setting of decentralized control with sub- sets of events observable (Σo,i)ni=1 and controllable(Σc,i)ni=1 by agent (supervisor)i and a specificationK⊆L=L(G).
We plunge the decentralized control problem into the coordination control problem by setting
Ai=Σo,i and Ac,i=Σo,i∩Σc,i.
For simplicity, the same notation for projection is kept, that is, Pi :A∗→A∗i. The plant language G will be over- approximated by a two-level modular plant kni=1Pi(G), that
is, by the parallel composition of projections to events observable by local control agents. We will organize the local supervisors into a two-level hierarchy. Similarly as in mul- tilevel coordination control we will group the agents based on their interactions given by intersection of their alphabets, here observationsAi=Σo,i. The idea is to place agents with maximal shared observations to the same groups at the lowest level of the multilevel structure. Then, in an ideal situation, there will be no shared observations between different groups of agents, because all shared observations are realized within the low-level groups. This can be formalized by associating a square matrix with the number of shared events observed by both and try to find after permutation a block matrix structure such that the maximum of shared events is situated in the diagonal blocks, while off-diagonal blocks contain very small numbers (ideally zero matrices). Again, we denote m low- level groups of agents byIj, where j=1,2, . . .m, form≤n.
According to the two-level top-down architecture, we have to find an extension Ak (a high-level coordina- tor alphabet) of these ”high-level shared observations”
Ash=Sk6=`k,`∈{1,...,m}(AIk∩AI`) such that Ak⊇Ash and K= kmr=1PIr+k(K). Similarly, we have to extend the shared low- level observations in groups, i.e., Ash,r=Sk6=`k,`∈I
r(Ak∩A`),
forr=1,2, . . . ,m, to low-level coordinator alphabetsAk,j⊇ Ash,r such that PIr+k(K) =kj∈IrPj+kr+k(K), i.e., two-level conditional decomposability holds true.
It is a common assumption in both modular and co- ordination supervisory control that shared events have the same controllability status in all components, i.e.,Ai∩Ac,j⊆ Ac,i, for i,j=1,2, . . . ,n. Since Ai =Σo,i, it is clear that this assumption is equivalent to Σo,i∩Σc,j⊆Σc,i, for i,j= 1,2, . . . ,n. Hence, by Theorem 18, separability implies coob- servability and the condition stated therein is not restrictive.
More precisely we have the following.
Lemma 19: If Σo,i∩Σc,j⊆Σc,i, for i,j=1,2, . . . ,n, then (Ai)ni=1 and (Ac,i)ni=1 defined above satisfy Ai∩Ac,j⊆Ac,i, for alli,j=1,2, . . . ,n.
OnceAk andAkj, for j=1,2, . . . ,m, are found such that two-level conditional decomposability holds, we compute languages Lkj =L(Gkj), sup Ck
j, and sup Ci+k
j as in the previous section, cf. formula (2), where L(Gi)are replaced by Pi(kni=1L(Gi)). Namely, for all j =1,2, . . . ,m, Lkj = kni=1Pkj(Pi(L)), sup Ck
j=sup C(Pkj(K),Lkj,Akj,u), and for all i∈Ij, sup Ci+k
j =sup C(Pi+kj(K),Pi(L)ksup Ck
j,Ai+kj,u).
Theorem 20: LetK⊆Lbe languages, and let K be two- level conditionally decomposable with respect to (Ai)ni=1, Ak,j, and Ak. Then kmj=1ki∈Ij sup Ci+k
j is a sublanguage of K controllable with respect to L and Au, and coobservable with respect toL and(Ai+kj)j=1,...,m,i∈Ij.
If kmj=1∩i∈IjPki+kj
j (sup Ci+k
j) is controllable with respect to L(Gkj) and Akj,u, and for all j=1,2, . . . ,m, Pkkj is an Lkj-observer and OCC for Lkj, then kmj=1ki∈Ijsup Ci+k
j =
sup 2cC(K,L,Ai+kj)is the largest controllable and coobserv- able language we can obtain using the two-level coordina- tion.
Note that according to the above results there are stronger sufficient conditions that are more suitable for verifica- tion, namely that for j = 1,2, . . . ,m and i ∈ Ij, either Pki+kj
j (sup Ci+k
j) = sup Ck
j, or Pki+kj
j are (Pii+kj)−1Pi(L)-
observers and OCC for (Pii+kj)−1Pi(L). The first condition has the advantage that if it holds, our results can be extended to the non-prefix-closed case. The second condition has the advantage that there are known algorithms to extend the local alphabets such that corresponding projections become OCC and satisfy the observer property. Finally, let us point out that without any assumption (except two-level conditional decomposability)Mmight be a too small language, because without the above additional conditions we cannot guaran- tee the optimality with respect to the coordination control (sup 2cC(K,L,Ai+kj)) and the optimality is lost twice (it is potentially lost due to over-approximation ofLbykni=1Pi(L) unless Lis decomposable).
C. Example
Let the languages K and L be given by generators on Figs. 1 and 2, respectively. The alphabets of local agents are Σo,1={a,b,u1,u}, Σo,2={a,b,u2,u}, Σo,3={v,b,v1,b1}, Σo,4={v,b,v2,b2}, Σc,1=Σc,2={a,b}, Σc,3={v,v1,b1}, Σc,4={v,v2,b2},Σu={b,u,u1,u2}, andΣc=Σ\Σu.
Note thatK is not controllable with respect toL, because e.g. v2v1b∈KΣu∩L, but v2v1b6∈K. On the other hand,K is coobservable with respect to L andΣo,i, for i=1,2,3,4.
However, N=sup C(K,L,Σu) is not coobservable with re- spect toLandΣo,i, fori=1,2,3,4, anymore. This is because bothv1v2∈Landv2v1∈L,v1v2∈N,v2∈N, whilev2v16∈N.
It is clear that at least one of the agents 3 and 4 has to observe both v1 and v2 in order to issue a correct control decision. This means that it is not clear how to compute a sublanguage that is at the same time controllable and coobservable. Also notice that if b were controllable, K would be controllable, but still not coobservable. We now use our two-level coordination control approach.
First, we project the plant to the alphabets of observable events, cf. Fig. 3. The inclusionL⊆ kni=1Pi(L)is strict. We set Ai=Σo,ifori=1,2,3,4. Now we have to extendAsh= (AI1∩ AI2) = (A1∪A2)∩(A3∪A4) ={b}such that forAk⊇Ash, and K=P1+2+k(K)kP3+4+k(K). It turns out that no extension of Ash is needed, i.e., Ak={b}.
We now need to find low-level coordination alphabetsAk1 andAk2 such that conditions on the low-level for two-level conditionally decomposability hold true. SinceP1+2+k(K) = P1(K)kP2(K), there is no need to extend Ash,1=A1∩A2= {a,u,b} and we take Ak1 =Ash,1. Therefore, P1+2+k(K) = P1+k1(K)kP2+k1(K), with Ak1 ={a,u,b}. In particular, co- ordinator Lk1 is not needed. Secondly, P3+4+k(K) is not decomposable with respect to A3+k and A4+k. Hence, we have to extendAkbyAk2 to make the equationP3+4+k(K) = P3+k2(K)kP4+k2(K)hold true. We chooseAk2={v1,b2,v,b}
and it can be checked that P3+4+k(K) is conditionally de- composable with respect to alphabets A3,A4, andAk2. The
Fig. 1. SpecificationK
Fig. 2. PlantL
corresponding coordinator for the second group is Lk2 = kni=1Pi(L) ={ε,v,vb2,v1,v1b}.
Now starts the actual computation of a sublanguage of K that is both controllable with respect to L and Au and coobservable with respect to L and observable al- phabets A1+k1,o, A2+k1,o, A3+k2,o, A4+k2,o. Note that K is not two-level conditionally controllable, hence the con- structive procedure is needed. In fact, P4+k2(K) given in Fig. 4 is not controllable with respect to P4(L)kPk2(K) given in Fig. 5. Therefore, corresponding supervisor S4
for P4(L) is not simply given by P4+k2(K), but has to be computed. Its language is given by sup C4+k
2 =
sup C(P4+k2(K),P4(L)ksup Ck
2,Au,4+k2). First, we compute the supervisor sup Ck
2 for coordinatorLk2. We have sup Ck
2=
sup C(Pk2(K),Lk2,Au,k2) = Pk2(K) = Lk2 computed above.
Thus, sup C4+k
2 ={v1,v2,b,v2,vb2} is given by Fig. 5.
This supervisor simply disables v1 after v2 has occurred.
b
u1 a
u b
u2 a
u v1
b v
b1
v2
b v
b2
Fig. 3. ProjectionsP1(L), . . . ,P4(L)
v1
b v
b1
b2
b2
v1
v2
v2
v1
b v
b2
Fig. 4. P3+k2(K)andP4+k+k2(K)
Similarly, we have to derive the supervisor sup C3+k
2 for P3(L). We note thatP3+k2(K), given in Fig. 4, is controllable with respect to P3(L)kPk2(K). Thus, sup C3+k
2 =P3+k2(K).
Concerning the first group of agents, no control action is required for supervisors sup C1+k
1 and sup C2+k
1, because P1+2(K) =P1(L)kP2(L). Moreover, sup C1+k
1 =P1+k1(K) = P1(K) =P1(L)and sup C2+k
1=P2+k1(K) =P2(K) =P2(L).
Since all conditions in Theorem 14 are satisfied, it holds true that sup C1+k
1ksup C2+k
1ksup C3+k
2ksup C4+k
2 =
sup 2cC(K,L,Ai+kj). In particular, it is controllable with respect toL andAuand coobservable with respect to L and A1+k1,o,A2+k1,o,A3+k2,o,A4+k2,o. /
V. CONCLUDING REMARKS
We have extended existential results of multilevel coor- dination control to constructive results with a procedure to compute the supremal sublanguage achievable in the two- level architecture. The constructive results have been applied to decentralized supervisory control with communicating supervisors, where local supervisors communicate with each other via a coordinator of the group.
VI. ACKNOWLEDGMENTS
The authors would like to thank St´ephane Lafortune and Feng Lin for a very fruitful discussion during their visit at the University of Michigan and Wayne State University.
The research was supported by the M ˇSMT grant LH13012 (MUSIC), by RVO: 67985840, and partially by the DFG in grant KR 4381/1-1.
v1
v2
v2
v1
b v
b2
v1
v2
v2
b v
b2
Fig. 5. L4kPk2(K)and sup C4+k2
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